1D Flux Power Spectrum (P1D) Overview

• 1D Flux Power Spectrum (P1D) is the line-of-sight power spectrum derived from Lyman-α forest flux fluctuations, encapsulating intergalactic medium characteristics.
• It is calculated by normalizing transmitted flux and applying Fourier transforms to model effects like thermal broadening, redshift-space distortions, and metal contamination.
• P1D analysis informs cosmological parameters and astrophysical feedback by comparing synthetic simulations with observational spectroscopic data.

The one-dimensional flux power spectrum, usually written $P_{1\mathrm{D}}$ or $P_{\mathrm{1D}}$, is the line-of-sight power spectrum of Lyman-$\alpha$ forest transmitted-flux fluctuations measured in quasar spectra. It is constructed from the transmitted flux $F=e^{-\tau}$ and the mean-normalized fluctuation field $\delta_F = F/\langle F\rangle - 1$, and it compresses the clustering of intergalactic neutral hydrogen into a function of redshift and line-of-sight wavenumber. Because the forest responds to density, ionization state, temperature, peculiar velocities, and thermal broadening, $P_{1\mathrm{D}}$ is used simultaneously as a probe of small-scale matter clustering, the thermal and ionization history of the intergalactic medium (IGM), reionization, dark matter microphysics, neutrino mass, and a wide class of astrophysical and instrumental systematics (Cabayol-Garcia et al., 2023).

1. Formal definition and mathematical structure

In its standard form, the transmitted flux along a line of sight is written

$F(v)=e^{-\tau(v)},$

where $\tau$ is the total Lyman-$\alpha$ optical depth evaluated in velocity or logarithmic-wavelength space. The fluctuation field is then

$$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$

and the 1D flux power spectrum is

$P_{\mathrm{1D}}$0

This is the quantity directly estimated from observed or synthetic spectra (Karaçaylı et al., 2023).

The line-of-sight wavenumber is usually reported in velocity units, $P_{\mathrm{1D}}$1 in $P_{\mathrm{1D}}$2, because the spectra are naturally sampled on logarithmic wavelength grids or equivalent velocity coordinates. A commonly used conversion to comoving units is

$P_{\mathrm{1D}}$3

or, equivalently,

$P_{\mathrm{1D}}$4

which is required when comparing observational measurements to cosmological simulations and emulators (Ho et al., 22 Sep 2025).

Formally, $P_{\mathrm{1D}}$5 is related to the 3D flux power through projection over transverse modes. In anisotropic redshift space,

$P_{\mathrm{1D}}$6

with $P_{\mathrm{1D}}$7 and $P_{\mathrm{1D}}$8. For an isotropic field, this reduces to the familiar expression

$P_{\mathrm{1D}}$9

However, for the Lyman-$\alpha$0 flux this isotropic projection is not itself a sufficient physical model, because the flux is a nonlinear transform of the density field and is further modified by thermal broadening and redshift-space distortions (Coughlin et al., 2018).

A persistent misconception is therefore that $\alpha$1 is simply the 1D matter power spectrum measured through neutral hydrogen. The literature consistently treats this as incomplete. The observable is instead the power spectrum of a nonlinear, thermodynamically modulated, redshift-space flux field, and its interpretation requires either hydrodynamical forward modeling or an emulator calibrated on such simulations (Ridkokasha et al., 20 May 2025).

2. Physical origin of the signal

The flux field is set by the optical depth, and the optical depth is built from the neutral hydrogen density, temperature, and line-of-sight velocity field, including both peculiar velocities and thermal broadening. In simulation pipelines based on particle or mesh data, the relevant quantities are interpolated onto skewers, the neutral fraction is obtained from ionization equilibrium or simulation chemistry, and the optical depth is assembled with explicit Doppler broadening and redshift-space mapping (Coughlin et al., 2018).

A widely used organizing approximation is the fluctuating Gunn–Peterson picture, in which

$\alpha$2

with

$\alpha$3

Within this framework, $\alpha$4 is shaped by the amplitude and slope of the underlying matter power spectrum, the mean transmitted flux, the temperature-density relation, thermal broadening, pressure smoothing, and peculiar-velocity distortions (Cabayol-Garcia et al., 2023).

These physical dependencies are scale dependent. Thermal broadening suppresses high-$\alpha$5 power by smoothing individual absorbers in velocity space. Pressure smoothing suppresses baryonic fluctuations on small physical scales through the integrated thermal history. Redshift-space distortions encode the line-of-sight velocity field. The mean flux, or equivalently $\alpha$6, rescales the overall absorption level and strongly modulates the power spectrum amplitude (Cabayol-Garcia et al., 2023).

Reionization affects $\alpha$7 in a particularly structured way. In the CROC simulations, the shape of the flux power spectrum over $\alpha$8 is found to be rather insensitive to the timing of reionization, while the amplitude evolves rapidly and is almost perfectly correlated with the timing of reionization. In that analysis, the amplitude summary statistic

$\alpha$9

tracks the redshift of overlap of ionized bubbles much more strongly than the detailed spectral shape (Mishra et al., 2021). This suggests that, for late-reionization applications, amplitude evolution and shape evolution need not carry the same information.

Astrophysical feedback modifies the same observable through the thermal and ionization state of the IGM. In Simba-based hydrodynamic simulations, AGN jet feedback suppresses the Lyman-$F=e^{-\tau}$0 forest $F=e^{-\tau}$1, especially at low redshift and high $F=e^{-\tau}$2, by heating and ionizing neutral gas and by thermally broadening absorbers. At $F=e^{-\tau}$3, the effect is reported as about $F=e^{-\tau}$4 for $F=e^{-\tau}$5 and about $F=e^{-\tau}$6 for $F=e^{-\tau}$7, whereas below $F=e^{-\tau}$8 the changes become much larger (Chen et al., 2024).

3. Estimation from spectra, simulations, and quadratic estimators

In simulation work, $F=e^{-\tau}$9 is usually measured from large ensembles of synthetic skewers. Examples include post-processed dark-matter or hydrodynamical simulations in which optical depths are constructed from density, temperature, and velocity fields, and synthetic spectra are extracted along random lines of sight. Typical implementations generate thousands of skewers per snapshot and then estimate the power by Fourier transforming $\delta_F = F/\langle F\rangle - 1$0 and averaging $\delta_F = F/\langle F\rangle - 1$1 over the skewer ensemble (Pirecki et al., 22 Sep 2025).

The statistical estimation problem in survey data is more complicated because spectra are masked, noisy, heterogeneous in resolution, and affected by continuum uncertainty. Two estimator classes dominate current work. The first is the fast Fourier transform estimator, which is operationally simple and efficient but requires explicit treatment of masking windows, instrumental deconvolution, and noise subtraction. The second is the optimal quadratic or quadratic maximum-likelihood estimator, which works in pixel space, builds the covariance as $\delta_F = F/\langle F\rangle - 1$2, and naturally incorporates inverse-variance weighting, spectrograph resolution matrices, masking, and continuum-mode marginalization (Karaçaylı et al., 2020).

In its standard form, the quadratic estimator solves for bandpowers $\delta_F = F/\langle F\rangle - 1$3 through quadratic forms in the data vector and the covariance response matrices. A representative expression is

$\delta_F = F/\langle F\rangle - 1$4

with Fisher matrix

$\delta_F = F/\langle F\rangle - 1$5

In Lyman-$\delta_F = F/\langle F\rangle - 1$6 applications, this formulation is used precisely because it is robust to irregular sampling, gaps, time evolution within a forest, and continuum errors (Karaçaylı et al., 2023).

Recent DESI work has also introduced cross-exposure variants of the quadratic estimator. By cross-correlating independent exposures of the same quasar rather than auto-correlating coadded spectra, these estimators eliminate the need to model the pipeline noise power in the mean, because uncorrelated noise does not bias the cross-spectrum. This development is explicitly presented as a route to stronger control of one of the dominant survey systematics (Karaçaylı et al., 12 May 2025).

Estimator validation has become an essential part of $\delta_F = F/\langle F\rangle - 1$7 analysis. In DESI DR1, both the optimal estimator and the FFT estimator were tested on synthetic 1D realizations and large CCD image simulation suites. After applying masking and continuum-bias corrections, both the bandpowers and their covariances were found to be unbiased in Kolmogorov–Smirnov tests, while the highest-$\delta_F = F/\langle F\rangle - 1$8 regime remained limited by a few-percent resolution error budget rather than by estimator failure (Karaçaylı et al., 16 Sep 2025).

4. Survey measurements and empirical dynamic range

Modern $\delta_F = F/\langle F\rangle - 1$9 measurements span both intermediate-resolution, very large samples and high-resolution, smaller samples. The SDSS DR14 analysis measured the 1D Ly$P_{1\mathrm{D}}$0 forest flux power spectrum from 43,751 quasars selected from a parent visually inspected sample of 180,413 spectra, yielding 94,558 forest segments. That measurement covered thirteen redshift bins from $P_{1\mathrm{D}}$1 to $P_{1\mathrm{D}}$2 and scales up to $P_{1\mathrm{D}}$3, with statistical uncertainties about a factor of two smaller than the previous DR9 result (Chabanier et al., 2018).

DESI early data extended the same program with a quadratic maximum-likelihood analysis of 54,600 quasars after BAL removal, using $P_{1\mathrm{D}}$4 bins from $P_{1\mathrm{D}}$5 to $P_{1\mathrm{D}}$6. That study emphasized detailed treatments of noise calibration, spectrograph resolution, side-band subtraction, DLA masking, and continuum marginalization, and reported that the DESI $P_{1\mathrm{D}}$7 was higher than eBOSS by about $P_{1\mathrm{D}}$8 to $P_{1\mathrm{D}}$9 across redshift, with the largest sensitivity to systematics in the $F(v)=e^{-\tau(v)},$0 bin (Karaçaylı et al., 2023).

DESI DR1 pushed the sample size substantially further. In the optimal-estimator measurement, the Ly$F(v)=e^{-\tau(v)},$1 forest sample used for $F(v)=e^{-\tau(v)},$2 contains 314,241 quasars in the rest-frame interval $F(v)=e^{-\tau(v)},$3–$F(v)=e^{-\tau(v)},$4 \AA. The recommended analysis cuts are $F(v)=e^{-\tau(v)},$5 and $F(v)=e^{-\tau(v)},$6, with

$F(v)=e^{-\tau(v)},$7

The same DR1 release also produced an FFT-based measurement, and the two pipelines were reported to be consistent with each other while constituting the most precise intermediate-resolution $F(v)=e^{-\tau(v)},$8 measurement to date (Karaçaylı et al., 12 May 2025).

On the high-resolution side, the KODIAQ, SQUAD, and XQ-100 analysis used an optimal quadratic estimator on 538 quasars and measured $F(v)=e^{-\tau(v)},$9 over $\tau$0 and $\tau$1. These data provide direct access to the small-scale modes $\tau$2 that are inaccessible to SDSS- or DESI-like resolution, and therefore sharpen sensitivity to thermal cutoffs and dark matter free-streaming (Karaçaylı et al., 2021).

Taken together, these measurements define a hierarchy of observational regimes. SDSS and DESI deliver very large samples with stringent control of survey systematics on moderate $\tau$3. High-resolution spectroscopy reaches substantially smaller scales with much smaller sample sizes and more acute sensitivity to absorber selection, line contamination, and thermal-history assumptions. This division of labor is explicit in current cosmological analyses (Ho et al., 22 Sep 2025).

5. Emulators, compressed inference, and cosmological use

Because direct likelihood evaluation with hydrodynamical simulations is computationally prohibitive, much of the modern literature uses emulators. One influential strategy is to emulate $\tau$4 as a function not of a full cosmological parameter vector but of summary variables that directly control the small-scale linear matter power: an amplitude and a local slope at a pivot scale, combined with instantaneous IGM parameters such as $\tau$5, $\tau$6, $\tau$7, and $\tau$8. The LaCE neural-network emulator implements this approach with a mixture-density network trained on 60 MP-Gadget hydrodynamical simulations and achieves sub-percent precision over $\tau$9–$\alpha$0 and $\alpha$1–$\alpha$2, while generalizing to massive neutrinos, running of the spectral index, and curvature not present in the training set (Cabayol-Garcia et al., 2023).

A complementary route is Gaussian-process emulation on high-resolution simulation suites. The Lyssa program uses 18 Nyx simulations with $\alpha$3 hydrodynamical cells in a $\alpha$4 comoving box and constructs a GP emulator within the lym1d likelihood framework. In that formulation, the cosmological information is compressed into the linear matter power amplitude and slope parameters $\alpha$5 and $\alpha$6 near $\alpha$7 at $\alpha$8, while the IGM is described by $\alpha$9, $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$0, $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$1, and $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$2 (Lalengmawia et al., 2024).

DESI DR1 has already used this emulator-based logic in a direct cosmological analysis. In a compressed-parameter framework, the DESI DR1 1D flux power spectrum constrains the amplitude and logarithmic slope of the linear matter power spectrum at $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$3 and $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$4, yielding

$$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$5

When combined with DESI BAO and CMB measurements from Planck, ACT, and SPT-3G, these data sharpen constraints on $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$6, the running $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$7, and the running of the running $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$8, even though they do not significantly tighten the combined limit on the sum of neutrino masses (Chaves-Montero et al., 29 Jan 2026).

High-resolution cosmology analyses use related but distinct emulation frameworks. The PRIYA emulator, based on multi-fidelity MP-Gadget simulations with inhomogeneous He II reionization, predicts $$\delta_F(v,z)=\frac{F(v,z)}{\langle F\rangle(z)}-1,$$9 over $P_{\mathrm{1D}}$00–$P_{\mathrm{1D}}$01 and $P_{\mathrm{1D}}$02–$P_{\mathrm{1D}}$03. Applied to XQ100 and KODIAQ-SQUAD, it shows that XQ100 alone yields $P_{\mathrm{1D}}$04 constraints consistent with eBOSS and Planck, whereas KODIAQ-SQUAD can be biased high in $P_{\mathrm{1D}}$05 because of selection bias toward high-column-density absorbers (Ho et al., 22 Sep 2025).

Likelihood-free approaches have also entered the field. Using CAMELS cosmological hydrodynamic simulations, simulation-based inference with a normalizing flow has been shown to recover unbiased posteriors for $P_{\mathrm{1D}}$06 and $P_{\mathrm{1D}}$07 when training and testing are performed on the same baryonic model, while training on one galaxy-formation model and testing on another induces a positive bias of about $P_{\mathrm{1D}}$08 in $P_{\mathrm{1D}}$09. Multi-domain training on both baryonic models removes that bias, illustrating that the mapping from cosmology to $P_{\mathrm{1D}}$10 is now precise enough that baryonic domain shift itself becomes an inference problem (Sinigaglia et al., 13 Mar 2026).

6. Contaminants, degeneracies, and current interpretive limits

The dominant limitations in $P_{\mathrm{1D}}$11 analysis are not purely statistical. High-column-density absorbers, metal lines, continuum fitting, spectral resolution, noise calibration, mean-flux uncertainty, UV-background fluctuations, thermal-history uncertainty, and baryonic feedback all reshape the observable in ways that overlap cosmological signatures. This is why nearly every modern analysis treats nuisance modeling as part of the primary forward model rather than as a peripheral correction (Chaves-Montero et al., 29 Jan 2026).

High-column-density absorbers are a canonical example. In Illustris-based simulations, LLSs, sub-DLAs, and DLAs bias the 1D flux power through damping wings that correlate absorption away from the absorber center itself. Their effect is naturally described as a multiplicative bias on the forest-only power, and column-density-dependent templates have been provided precisely so that analyses can model residual contamination after clipping the largest wings (Rogers et al., 2017).

Metals introduce both broad-band and oscillatory structure. In DESI DR1 cosmological modeling, residual metal contamination overlapping the forest is represented by multiplicative Si III and Si II oscillation terms, together with additive same-ion templates such as Si II–Si II and doublet residuals for Mg II and C IV. The dominant Ly$P_{\mathrm{1D}}$12–Si III feature corresponds to a velocity separation near $P_{\mathrm{1D}}$13, and the Ly$P_{\mathrm{1D}}$14–Si II feature near $P_{\mathrm{1D}}$15, which generate coherent oscillations in $P_{\mathrm{1D}}$16 if not explicitly modeled (Chaves-Montero et al., 29 Jan 2026).

Mean transmission is another central degeneracy. In the Lyssa-based eBOSS analysis, the apparent low value of the linear power amplitude parameter $P_{\mathrm{1D}}$17 relative to Planck is driven by its correlation with the mean transmission of the forest. Without an external prior, the DR14 fit gives $P_{\mathrm{1D}}$18 at $P_{\mathrm{1D}}$19 confidence, whereas a well-motivated prior on the mean transmission yields $P_{\mathrm{1D}}$20, removing the tension with Planck (Lalengmawia et al., 2024). This is a direct demonstration that $P_{\mathrm{1D}}$21 cosmology is only as robust as its treatment of $P_{\mathrm{1D}}$22.

Selection effects can dominate high-resolution analyses. The PRIYA study finds that KODIAQ-SQUAD favors a significantly higher $P_{\mathrm{1D}}$23 than eBOSS or Planck, and attributes this to a selection bias toward high-column-density absorbers, especially Lyman limit systems. When the analysis is restricted to $P_{\mathrm{1D}}$24–$P_{\mathrm{1D}}$25 and $P_{\mathrm{1D}}$26, the inferred cosmological parameters become consistent with eBOSS and XQ100, while the strong LLS preference weakens (Ho et al., 22 Sep 2025). This suggests that cosmology and thermal nuisance parameters are largely sensitive to different scales, but only after absorber contamination is controlled.

The literature also documents cases in which $P_{\mathrm{1D}}$27 is intrinsically insensitive to a target parameter. In simulations probing the CPL form of time-dependent dark energy,

$P_{\mathrm{1D}}$28

the Lyman-$P_{\mathrm{1D}}$29 forest flux power spectra of Planck-allowed models are visually and statistically indistinguishable from $P_{\mathrm{1D}}$30CDM in Anderson–Darling tests, with only a marginal effect even for extreme models. The conclusion is that the intrinsic variance of $P_{\mathrm{1D}}$31, together with IGM and mean-flux degeneracies, dominates over the signal expected from allowed time-dependent dark energy models (Coughlin et al., 2018).

A comparable caution applies to reionization timing. CROC simulations show that the amplitude of the high-redshift flux power spectrum is strongly correlated with the redshift of overlap, whereas the shape is rather insensitive to reionization timing, in explicit disagreement with earlier claims of large shape sensitivity from other radiative-transfer calculations (Mishra et al., 2021). This is not merely a numerical detail: it determines whether the informative statistic is a scale-dependent distortion or an amplitude-redshift offset.

The present status of $P_{\mathrm{1D}}$32 can therefore be stated precisely. It is a mature precision observable, with validated estimators, survey-scale data sets, sub-percent emulators, and direct cosmological constraints already in hand. At the same time, its interpretation is inseparable from forward models for mean transmission, thermal history, pressure smoothing, metals, high-column-density absorbers, resolution, noise, and feedback. This suggests that the future gains from DESI and high-resolution spectroscopy will come as much from better control of these couplings as from raw increases in sample size.